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 Incoming links: Causality

Damped ferromagnetic spin wave Created 2026-09-28 Updated 2026-10-07

A damped ferromagnetic spin wave has a complex pole

ω = \pm ω_{k} - i Γ_{k}

. Both poles of its retarded Green function must lie in the lower half of the complex frequency plane so that the response respects causality.

J^{μ}

: every source volume element contributes with the inverse-distance Green-function factor. Electromagnetic influence propagates at speed

c

, so causality requires the source to be evaluated where its worldline intersects the field event's past light cone. Thus

A^{μ} (x, t) = \frac{μ _{0}}{4 π} \int_{R^{3}} \frac{J ^{μ} ( x ^{'} , t - ∣ x - x ^{'} ∣/ c )}{∣ x - x ^{'} ∣} d^{3} x^{'} .

The source time is exactly the retarded time

t_{ret} = t - ∣ x - x^{'} ∣/ c

, rather than the observation time

t

For a damped ferromagnetic spin wave, shifting both poles into the lower half-plane preserves causality:

G_{R} (ω, k) = \frac{ω _{k} / s}{( ω + i Γ _{k} ) ^{2} - ω _{k}^{2}} .

ϱ (ω, k) = \frac{4 ( ω _{k} / s ) ω Γ _{k}}{( ω ^{2} - ω _{k}^{2} - Γ _{k}^{2} ) ^{2} + 4 ω ^{2} Γ _{k}^{2}} .

S (ω, k) = \frac{4 T ( ω _{k} / s ) Γ _{k}}{( ω ^{2} - ω _{k}^{2} - Γ _{k}^{2} ) ^{2} + 4 ω ^{2} Γ _{k}^{2}} .

S (k) = \frac{4 T ω _{k} Γ _{k}}{s ( ω _{k}^{2} + Γ _{k}^{2} ) ^{2}},

which has the stated proportionality after absorbing the normalization

4/ s